analysis of interference factors of air turbine cascades

Engineering Applications of Computational Fluid Mechanics Vol. 7, No. 4, pp. 496–506 (2013)

Received: 23 Nov. 2012; Revised: 12 Jul. 2013; Accepted: 5 Aug. 2013

496

ANALYSIS OF INTERFERENCE FACTORS OF AIR TURBINE

CASCADES

Andrei Gareev, Buyung Kosasih * and Paul Cooper

School of Mechanical, Materials and Mechatronics Engineering, University of Wollongong,

Northfields Avenue, NSW 2522 Australia

* E-Mail: [email protected] (Corresponding Author)

ABSTRACT: This paper reports viscous compressible aerodynamics analysis of two-dimensional cascades (arrays

of aerofoils). The study elucidates the effect of cascade parameters such as solidity, and angle of attack, on the

interference between blades. Understanding the interference behaviour is important in the performance analysis of

air turbines such as Wells turbine commonly used in oscillating water column (OWC) - wave energy converter

(WEC) system. The present analysis shows that isolated two-dimensional lift coefficient modified by interference

factors, ko, based on inviscid flow theory is acceptable for low solidity cascades, i.e. 0.5, and that stall occurs at

lower angle of attack in cascade compared to isolated aerofoil. When > 0.5, ko is shown to depend on both and

especially in the post-stall flow. The result suggests that interference factors that is applicable for wide range of flow

conditions as in Wells turbines should also include the effect in the modifier.

Keywords: aerofoils, cascade, computational fluid dynamics, interference factor, solidity, Wells turbine

1. INTRODUCTION

Wide variety of technologies have been proposed

to extract energy from ocean waves ranging from

floating structures that flex and recover energy

through hydraulic systems such as the Pelamis

wave farm located off the coast of Portugal

(Pelamis, 2012) through to various types of buoy

tethered to ocean floor such as the OPT's

PowerBuoy (Ocean Power Technologies, 2012).

Oscillating Water Column (OWC) system shown

in Fig. 1a is one of the most promising wave

energy technologies. Full scale examples of such

devices are the Limpet on the Isle of Islay in the

UK (Whittaker et al., 2003), and the Pico plant in

the Azores (Brito-Melo et al., 2008).

Most oscillating water column wave energy

converters (OWC-WECs) use axial flow turbines

(Fig. 1b) to harness the energy of the oscillating

variable velocity air as the free-surface of the

water rises and falls inside the OWC chamber.

Because of these conditions, the turbines must be

capable of effectively harnessing the pneumatic

energy generated over a wide range of angles of

attack. In the first-level design analysis of OWC-

WEC turbines, such as Wells turbine (Figs. 1b

and c), the work done by the fluid on the blades is

calculated (Gato and Falcao, 1984) in a way

similar to BEM approach. Despite of the relative

simplicity of the approach, it relies on accurate

lift and drag data. Because the flow between

adjacent blades in cylindrical or linear series of

blades (cascade) can be quite different to that over

isolated aerofoils, the cascade lift and drag are

different to those of isolated aerofoil due to the

“interference” that each blade has on the flow

field around its neighbours. For OWC-WEC

Wells turbines, which often have high solidity

rotors, the interference effect between the blades

in circular cascade can be even more significant.

The parameters affecting the degree of the mutual

interference shown in Fig. 2 include (Weinig,

1964): the solidity of the cascade, (= c/s), angle

of attack, , and the blade stagger angle, . The effect of the interference between blades in a

cascade is usually expressed in terms of an

interference factor, ko, which is the ratio of the lift

coefficient of a blade in a cascade, CL, to that of

an isolated blade, CL0. The interference of

adjacent blades at various stagger angles has been

analytically calculated using inviscid potential

flow analysis such as the Weinig’s model of

linear cascade of flat or curved plates (Weinig,

1964) and aerofoil surfaces as singularities

(Raghunathan, 1995 and 1996). These results are

expected to be valid in the stall free flow

condition.

Weinig’s formula has been used by several

researchers (Gato and Falcao, 1988 and 1984;

Raghunathan and Beattie, 1996) in OWC-WEC

turbines analyses. This is acceptable if the turbine

operates in stall free conditions. However Wells

turbine is well known to have very narrow high

efficiency operating condition which can be

Engineering Applications of Computational Fluid Mechanics Vol. 7, No. 4 (2013)

497

Fig. 1 Schematics of: (a) Oscillating Water Column

(OWC) Wave Energy Conversion (WEC)

device; (b) Wells turbine and (c) full-scale

Denniss-Auld turbine with variable pitch

angle blades (Alcorn and Finnigan, 2004).

Fig. 2 Cascade on axial flow turbine rotor with

stagger angle, °; chord length, c; blade

pitch, s; axial air velocity, VA; and tangential

velocity relative to aerofoils, VT. The angle of

attack and relative velocity are defined as m

= (1 + 2)/2 and Wm= (W1+W2)/2.

attributed to the operation of the turbines under

deep stall condition over significant portion of the

blades span. Deep stall starting from the tip

moving toward mid-span reduces the power

output significantly (Torresi et al., 2009). This

shows that the use of simple interference factor

based on inviscid and infinitesimally thin foils is

not suitable for Wells turbine analysis.

A number of approaches to reduce the loss and

increase cascade performance have been reported.

Sun et al. (2012) demonstrated the effect blade

leading profiles and wedge angle on turbine

cascades performance. Strong effect was shown

on the boundary layer separation and the trailing

loss. Constant chord Wells turbine is shown to

stall early compared to variable chord turbine

resulting in greater efficiency of variable chord

turbine by up to 60% (Govardhan and Chauhan,

2007). Guide vanes have also been introduced to

increase the tangential velocity at the rotor inlet

aiming at increasing turbine efficiency. Kumar

and Govardhan (2010) further studied the effect

of placing passive fence to reduce the secondary

flow losses.

The paper presents the results of CFD analyses of

the aerodynamics of linear cascades and discusses

the flow field through such cascades particularly

the effect of cascade parameters on the incipient

of stall and its implication on the blade lift and

drag. The merits and inadequacies of the previous

inviscid model of axial flow air turbines for

OWC-WECs such as that of Weinig (1964) are

also discussed.

2. CASCADE INTERFERENCE FACTOR

BASED ON INVISCID ANALYSIS

A simplified illustration of infinite linear cascade

with staggered blades is shown in Fig. 2. Because

of the blades interaction with each other, the lift

coefficient, CL, and the drag coefficient, CD, are

different to that of isolated aerofoil. Modifiers

expressed in terms of the ratio of the cascade lift

coefficient, CL, to that of isolated blade, CL0,

taken at the same angle of attack, known as the

interference factor (ko = CL/CL0) are commonly

adopted in Wells turbine analysis. Few

experiments to determine ko have been reported.

Raghunathan (1988) carried out experiment with

limited range of solidity and in free stall

condition, where it is shown that the Weinig’s

model is still applicable. As consequence, many

Wells turbine performance analysis still adopts

the interference factor based on linear cascade of

infinitesimally thin flat blades (Weinig, 1964). It

is expected that at high solidity and high angle of

attack the viscous and compressibility effects can

lead to different separation mechanism and hence

inception of stall. Additionally there is no report

on the post-stall aerodynamic behaviour of

cascade.

Weinig (1964) showed an exact solution for 2-

dimensional potential flow around row of flat

plates and the interference factor is independent

of attack angle, , and is only a function of blade

stagger angle, , and cascade solidity, . Using

the analytical formulation for the lift coefficient

for an isolated aerofoil prior to stall, Weinig

(1964) determined the lift coefficient for cascade

blade as follows

CL = 2ko sin(m) (1)

where m is the angle of attack based on the mean

of the flows upstream and downstream of the

cascade, and ko is a complex function of and s/c


498

(or 1/) shown in Fig. 3. For the special case with

= 0°, as in the case of a Wells turbine, the

interference factor (Weinig, 1964) is given by

2s

ctan

c

2s

ok (2)

Eq. (2) is widely accepted as the lift coefficient

modifier or interference factor of fixed-pitch ( =

0°) Wells turbine blades. CL modified by Eq. (2)

is shown together with experimental data for an

isolated NACA0012 aerofoil at Re = 7×105

(Sheldahl and Klimas, 1981) in Fig. 4. Solid

curve in this figure represents the cascade lift

coefficient calculated using the isolated aerofoil

data multiplied by the interference factor of Eq.

(2). In order to determine ko, and plot the cascade

lift curve, the following cascade dimensions: c =

0.1 m and s = 0.169 m were used in Eq. (2). It can

be clearly seen that the lift curve based on the lift

data modified by ko exhibits higher values of CL

than the isolated aerofoil lift coefficient and the

deviation seems to start from m=10o. On the

figure, CL calculated by CFD compressible model

(*) shows reasonable agreement at low angle of

attack (5o). However the CFD predicted CL at stall

and post-stall is much greater than the prediction

by Eq. (2). It is expected that the cascade

interference effect will be significant in the post

stall regime. The CFD approach discussed in

Section 3 was validated by comparing the

coefficient of lift of isolated aerofoil case. The

agreement between the experimental curve (dash

curve) and the numerical values () is very good

in the pre-stall region and reasonable in the post-

stall region.

Although Eq. (2) has been widely used in the

analysis of Wells turbines (Gato and Falcao,

1988; Raghunathan and Beattie, 1996) and

compared with experimental measurement of

Wells turbine effeiciency (Gato and Falcao, 1988;

Raghunathan et al., 1990) to the best of our

knowledge, there is still limited validation of the

relationship between ko and from practical

aerofoil cascades i.e. over large range of angle of

attack, and solidity in pre-stall and post stall

regimes and hence limited understanding of the

cascade effect in post-stall condition.

3. 2-DIMENSIONAL COMPUTATIONAL

FLUID DYNAMICS (CFD) MODEL

The present study used ANSYS CFX code to

simulate the flows around cascades of relevance

to OWC-WEC axial flow turbines under realistic

flow conditions i.e. viscous, turbulent flow and

Fig. 3 Interference factor, k0 = CL/CL0, from potential

flow theory (Weinig 1964).eff is the stagger

angle defined as eff = 90° - .

Fig. 4 Comparison of isolated experimental aerofoil

lift data CL0 by Sheldahl and Klimas (1981) at

Re = 7×105, corrected cascade lift data CL (by

k0 for s/c = 1.69, = 0.6) and present CFD

infinite cascade data.

over practical ranges of angle of attack and

Reynolds number. CFD interference factors were

derived and presented in terms of the effective

angle of attack based on the mean angles between

the aerofoil chord, and the upstream and the

downstream flow directions, m = (1+2)/2,

shown in Fig. 2.

2-dimensional, steady and compressible CFD

simulations over wide ranges of angles of attack

and cascade solidity were carried out. Grid

independence was verified for linear cascade

simulation and the number of unstructured

elements where coefficient of lift remains

unchanged with further refinement was of the

order of 1.5×106 (Gareev, 2011). A typical mesh

around the blade and type of boundary conditions

used are shown in Fig. 5. k- SST turbulence

model with automatic wall functions and the

-5 0 5 10 15 20 25-0.5

0

0.5

1

1.5

2

m

(degrees)

CL

CFD - infinite linear cascade

CFD - single airfoil

experimental curve of isolated aerofoil

modified experimental curve by Eq. (2)


499

Fig. 5 Typical mesh used in CFD simulations is shown in the upper figures. In general tetrahedral elements were

used and prismatic elements defined within inflation layers were used near the aerofoil surfaces.

Computational boundary conditions are shown in the lower figure: (1) upstream boundary – specified inlet

velocity; (2) upper and lower boundaries - periodic; (3) left and right side boundaries – free-slip wall; (4)

downstream boundary – opening with specified static pressure; and (5) airfoil surface – nonslip wall.

“high resolution” advection scheme and “auto

timescale” were used. As the flow can reach high

Mach number at the leading edge of the blades

where compressibility is significant total energy

model to account for the heat transfer was used.

The automatic near-wall treatment switches

between low-Re near wall formulation when y+ <

2 and otherwise standard wall function

formulation to resolve the flow close to the blade

surfaces. Inflation layers were used to create fine

mesh near the surface resulting in average y+

between 10 and 20. All simulations were carried

out at Re = 7×105 (based on chord length).

The computational domain and boundary

conditions are shown in Fig. 5. The computational

domain was bounded by inlet boundary with

specified velocity, turbulent intensity of 5% and

air temperature of 20o

C. The outlet boundary was

specified as opening held at atmospheric static

pressure. To model 2-dimensional flow only one

cell was used to represent domain thickness. Only

one aerofoil was used in the simulation with

periodic boundary as the interface between

adjacent blades.

4. RESULTS AND DISCUSSIONS

The lift coefficients from the present CFD

simulations at Re = 7×105 were first compared to

experimental isolated aerofoil data (Sheldahl and

Klimas, 1981) and to values corresponding to the

isolated experimental data modified by Eq. (2) in

Fig. 4. It is evident that the CFD predictions are in

good agreement with experimental lift data

corrected by applying Eq. (2) for ≤ 10º. This

confirms the validity range of Eq. (2). However in

the post-stall region, the CFD results show

significantly higher CL.

Interference factors for blades chord parallel to

the plane of the cascade ( = 0o) cases were

analysed for a range of solidities (1/ = s/c) and

mean angles of incidence, m. Results from these

simulations are presented in Fig. 6 together with

the interference factors calculated using Eq. (2).

It can be seen in Fig. 6 that the interference

factors obtained by Eq. (2) (Weinig 1964) and

viscous compressible CFD simulations match up

to 1/ = 2 or = 0.5 for all angles of attack up to

m = 20o. When > 0.5 the viscous and


500

Fig. 6 Lift coefficient interference factors, ko, for linear cascade ( = 0º) of NACA0012 aerofoils as function of 1/

and mean angle of incidence, m. The solid curve is based on Weinig inviscid flow analysis, Eq. (2). The

arrows indicate the stall attack angles at different solidities.

(a) m = 5° (b) m = 10°

(c) m = 15° (d) m = 20°

Fig. 7 Velocity vectors around cascade (NACA0012) of 1/ = 1.25 ( at different mean angles of incidence,

m.


501

(a) = 5° (b) = 10°

(c) = 15° (d) = 20°

Fig. 8 Velocity vectors around isolated aerofoil (NACA0012) at different mean angles of incidence, .

compressibility effects in general cause greater

lift than isolated aerofoil and the effects become

more pronounced at high angle of attack. It was

found that the difference widens in post-stall

region when m > 10o.

Figs. 7 and 8 show the velocity vectors around the

cascade blades and isolated aerofoil, respectively.

It can be seen that in cascade, stall occurs at lower

attack angle, namely m = 10o and m = 20o for

the isolated aerofoil. The existence of separation

bubble over the aerofoil suction side in cascade

results in lower lift and hence dip in the

interference factor (indicated by arrows in Fig. 7).

At m = 10o the flow is already in the post-stall

region in contrast to isolated aerofoil where stall

is not seen until about 20o as shown in Fig. 8. This

causes the lift to be lower than isolated aerofoil

lift. Similar phenomenon is also seen at = 0.6

where cascade stall occurs at m = 15o. The drop

(indicated by arrows in Fig. 7) in ko observed at

1/ = 1.25 ( = 0.8) and 1.667 ( = 0.6) at m =

10o and 15o can be explained by the effect of the

cascade in lowering stall angle. From this

observation it can be concluded that cascade

lowers stall angle, the greater the solidity the

lower the cascade stall angle.

As the angle of attack increases beyond m > 10o,

the difference between the Weinig interference

factor and the CFD calculated ko widens. This

indicates that cascade effect is more pronounced

at high solidity and in post-stall region.

The increase of ko at high angle of attack (m =

15o and 20o) seen in Fig. 6 can be attributed to

greater pressure difference between the pressure

and suction sides of the blades as shown in Figs.

10 and 11. This is not seen when m = 10o in Fig.

9. In fact there is a significant increase in pressure

on the suction side (circled in Fig. 9) due to early

stall correspond to Fig. 7b, leading to drop in lift.

The effects of cascade solidity on surface pressure

distribution are shown in Figs. 9 to 11. Few

observations can be made. Increased solidity

leads to higher pressure on the pressure sides at

m > 10° or in the post stall region. At high


502

Fig. 9 Surface pressure distributions on cascade blade for different solidities at m = 10°. The circle shows high

pressure in the separation region when = 0.8.

Fig. 10 Surface pressure distributions on cascade blade for different solidities at m = 15°.

solidity, , pressure distribution on the

blade surface is more uniform. It appears that

when solidityincreases, the proximity of blades

induces blockage effect which causes an increase

of the surface pressure on the pressure side and

decrease of the pressure on the suction side of the

cascade (see Figs. 10 and 11), creating larger

pressure difference between the suction and the

pressure sides. This explains the rapid increases

of ko when > 0.7 (s/c < 1.42).

Comparison of the streamlines around isolated

aerofoil with that around cascade shows two

interesting phenomena. Firstly, as stall is

approached with increasing angle of attack the

separation zones on the cascade blades start at the

leading edges of the blades where in isolated

aerofoil the flow remains attached downstream

from the leading edge, i.e. until the local adverse

pressure gradient causes the flow to detach.

Secondly, at higher cascade solidity, the

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-12

-10

-8

-6

-4

-2

0

2

4

normalised length

Cp

= 0.5

= 0.6

= 0.7

= 0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-10

-8

-6

-4

-2

0

2

4

6

8

normalised length

Cp

= 0.5

= 0.6

= 0.7

= 0.8


503

Fig. 11 Surface pressure distributions on cascade blade for different solidities at m = 20°.

(a) m = 5° (b) m = 10°

(c) m = 15° (d) m = 20°

Fig. 12 Streamlines around cascade blade (NACA0012) with obtained with compresible model at different

mean angles of incidence, m.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-15

-10

-5

0

5

10

15

20

normalised length

Cp

= 0.5

= 0.6

= 0.7

= 0.8


504

separation bubbles for the cascade contain paired

recirculation secondary flow zones, whilst in

isolated aerofoil the separation zone only contains

single recirculation cell. This suggests that the

mechanisms of separation zone formation in

cascades and isolated aerofoils are different. We

explore this issue in the following discussion.

As solidity increases, the mutual proximity of the

aerofoils results in very significant increases to

both the local velocity and the angle of attack at

the leading edges of the aerofoils (see the bold

arrows in Figs. 12c and d) in comparison to an

isolated aerofoil. This is more so in higher attack

angle (Figs. 12c and d) than in lower angle (Figs.

12a and b). The combined effect is thought to

initiate separation to occur closer to the leading

edge and to lead to larger separation zone with

paired recirculation cells. A comparison with

lower solidity cascade at < 0.5 demonstrates

that at lower solidity there is a relatively small

increase in velocity at the leading edge compared

to the superficial flow. Analysis of the flow fields

for other solidity cascades shows that starting

from =0.7, large increase of velocity results in

strong adverse pressure gradient that in turn

results in separation very close to the leading edge

of the aerofoil and a separation bubble over the

entire suction surface and consequently greater

drag and increased pressure on the pressure side

that lead to greater lift. The high velocity near the

leading edge (Mach > 1) as shown in Fig. 13

leads to lower leading edge pressure and overall

pressure over the blade suction side. The

important difference between the compressible

and incompressible models appears to be on the

onset of stall. Fig. 14 shows the streamlines

around the blades where stall is not seen until 15o

angle of attack where with compressible model

stall is seen at 10o

.

(a) m = 15° (b) m = 20°

Fig. 13 Mach number (NACA0012) in a cascade with at different mean angles of incidence, m.

(a) m = 10° (b) m = 15° (c) m = 20°

Fig. 14 Streamlines around cascade blade (NACA0012) obtained with incompresible model at different mean

angles of incidence, m.


505

The above analysis clearly demonstrates there is

strong dependence of interference factors on

solidity, stagger angle, angle of attack and

compressibility. The interference between blades

gives rise to high velocity in the blade spacing

and greater pressure difference between the

pressure and the suction sides.

5. CONCLUSIONS

Analytical lift interference factors for linear,

tandem cascades (with zero stagger angle, = 0°)

predicted by Weinig (1964) based on potential

flow methods have been compared with those

deduced from viscous compressible flow CFD

simulations. Weinig’s inviscid analysis showed

that lift interference factors, ko, for an infinite

cascade is independent of the angle of attack, ,

and dependent only on aerofoil stagger angle, ,

and cascade solidity, . Our CFD analyses have

provided confirmation that Weinig’s formulation,

Eq. (2) is adequate for linear cascades such as

NACA0012 for relatively low angles of attack

(m < 10°) prior to the incidence of stall.

However, at higher angles of attack (m > 10°) the

present analyses have shown that the interference

factor, ko, depends on both angle of attack and

solidity. This is shown to be due to a number of

factors including viscous and compressibility

effects that result in earlier flow separation and

stall for cascade compared to single aerofoil, and

the influence of the finite thickness of the

aerofoils, as compared to the infinitesimally thin

foils assume in inviscid flow analyses. Of

particular note are the effects of high cascade

solidity, which increases the local flow velocity,

and angle of attack at the vicinity of the leading

edges of the aerofoils, leading to changes in flow

separation and recirculation on the suction side of

the aerofoils as compared to an isolated aerofoil.

The issues dealt with in the present work may be

important in a wide range of flow situations,

including blade element modelling of turbines

under unsteady and/or reciprocating flow

conditions. Analysis of air turbines for Oscillating

Water Column Wave Energy Convertors (OWC-

WECs) is a case in point, where in practice the

OWC-WEC turbine aerofoils may operate under

stalled conditions for significant portions of the

reciprocating air flow process through the turbine.

Lastly it is noted that further work is still needed

to address the 3D effects due to the radial

variation of the solidity.

NOMENCLATURE

c aerofoil chord length

CD drag coefficient

CL lift coefficient

CL0 lift coefficient of isolated aerofoil

Cp

pressure coefficient, = 2

2

1W

pp r

ko lift interference factor, = CL/CL0

p pressure

pr reference pressure

Re Reynolds number, = c W/

s cascade pitch

u friction velocity, = /w

VA axial relative velocity

VT tangential relative velocity

W relative velocity

y+ Reynolds number, = u y/

angle of attack

angle of incidence

blade stagger angle

density

kinematic viscosity

solidity (= c/s)

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analysis of interference factors of air turbine cascades

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